# What is the relation between kernel functions, kernels used in convolution and null spaces of a matrix?

I have recently started learning about machine learning and have come across kernels and null spaces. I understand that null space is the set of all vectors that satisfy the equation A.v = 0 (Where A is a matrix). I have been taught that null space is a set of vectors that are squished to 0 when transformation matrix A is applied. Then I came across SVM where kernel functions are used. I read that null spaces are also called kernels of a matrix. My questions are as follows.

1. Are both kernel functions and null spaces same? If they are related, how are they related?
2. What is the relation between kernel functions used in SVM and null space of matrix? If yes, how? Is this derived from the null space of transformations used in SVM?
3. What is the reasoning behind kernels used in kernel convolution? For example: Is the Gaussian kernel used a representation of the transformation? How are they related to null spaces? If they are related, how are they related?

• Another motivation, for usage of the same word "kernel" in two contexts, might be from one of the first times a kernel function appeared in mathematics. An example, would be: Consider the inhomogeneous differential equation $y''+by'+cy=g(x)$ and let $K(x)$ be the solution to homogeneous version $y''+by'+cy=0$ (which in a sense is kernel, in the linear algebra sense). Then $\int K(t-s)g(s)ds$ is the solution to the inhomogeneous DE (in this integral $K$ is the kernel, as the weighting function, similar to SVM). – Hamed Jul 12 '18 at 15:28